The Android browser will, but the image would have to be online somewhere. There's no way that I've found to open a local file in the browser. I suppose you could create an HTML file with the image and view it that way, but it seems a lot of hassle for little reward.

Navigating to the folder the gif file(s) are in with another browser besides the stock Android browser works for me.

E.G.: if your gifs are in the Downloads folder, Open Firefox, go to address bar, type in: /storage/emulated/0/Download (it may auto-complete for you). You should get a basic Index of: ... page from which you can click on the name of the gif.

Tested with Firefox 21.0 and Chrome 27.0.1453.90 on a Nexus 4.
Both browsers accepted /storage/emulated/0/Downloads, but Chrome gave an Access Denied error on the shorter path /storage/emulated/0.

I recommend making a bookmark of the page, for easier access in future.